If the answer is shutting 6 Music, what exactly was the question?

Is £9m too much to spend on 6 Music's 695,000 listeners? Not when Radio 3 has 1.8 million listeners and £51m. Photograph: Graeme Robertson

I'm still waiting for the BBC to properly explain the decision to close 6 Music. I've heard plenty of soundbites, but I've yet to see their working out. I just want to see how the decision was arrived at, because everything offered so far either contradicts other decisions or fails to make sense.

Let's start with the latest musings of Caroline Thomson, the BBC's chief operating officer, who stated yesterday that 6 Music competes with commercial radio. "The average age of its listeners – 37 – is at the heart of the demographic targeted by commercial radio."

The logic implied by Thomson is that if you remove 6 Music from the listener's choice, you broaden the appeal of commercial radio.

Will listeners discover a commercial service to suit their tastes? Of course not, because commercial radio has nothing on the books like 6 Music. According to comparemyradio.com, 6 Music shares just 20% of its playlist with NME Radio, and just 7% per cent with Xfm. Nobody who currently listens to 6 Music will find a commercial service that fits.

So will commercial radio adapt their output to fill the vacuum? No. If there was money to be made, the commercial sector would be doing it – 6 Music's small audience size (695,000 weekly listeners) wouldn't have proved a deterrent if it was commercially viable to reproduce. 6 Music is too niche – there's little profit to be turned by aping its music format.

There's also the fact that 6 Music listeners are predominantly Radio 1 and Radio 2 listeners – 62% of 6 Music listeners choose to listen to one of the other – so the loss of 6 Music doesn't translate into increased listening for commercial radio, but for the BBC.

Thomson repeated the statement that the BBC is concerned about operating three popular music stations – Radio 1, Radio 2 and 6 Music.

So what's a "popular music station", exactly? Is it a station that plays popular music? As we've seen, 6 Music doesn't play popular music. Is it a popular station? Not with 695,000 listeners, it isn't – and the BBC agrees this point in the strategic review, talking about its "low reach and awareness".

Will it be a popular station in the future? More popular, probably – but again, there isn't a scrap of proof that it would damage commercial listening.

That leaves one last argument in the strategic review remaining – that 6 Music "delivers relatively few unique listeners to BBC radio". As mentioned, it's true there's significant overlap between the audience of 6 Music and those of other BBC services, but that argument is more than enough justification to close another digital service – Radio 7.

As the former executive producer of Radio 7, I don't like to pick on it; the service is adored by listeners and offers distinct output. But if you follow the BBC's logic for shelving 6 Music, how can you justify the continuation of Radio 7? According to the latest Rajar figures, 77% of its audience are Radio 4 listeners.

The BBC is super-serving Radio 4 listeners, not increasing listenership to BBC radio in any meaningful way. So why is it rebranding Radio 7 as Radio 4 Extra, a move that will actively discourage unique listeners? The remit of Radio 7 was to introduce a new audience to speech radio – listeners uncomfortable with the middle-class tone of Radio 4.

If budgets are the issue, it makes more sense to reinvent Radio 7 a listen-again service; none of the material broadcast is time-sensitive, so why does it need to be broadcast as linear radio? Why all the broadcast infrastructure and production, when you can invest in an online archive? How many people own a digital radio, but not a computer?

Nobody would be outraged if 6 Music had to cut its budget – £9m is a big purse for a music-based service. And nobody would argue with the statement that 6 Music has a relatively low number of listeners – though they might argue that Radio 2 has provided little support for its sister station, and that it's notoriously difficult to market a service as diverse as 6 Music.

But the BBC cannot prove in any meaningful way that 6 Music threatens commercial listening now, or will do so in the future. It can't argue that it's operating three popular music stations when one of them clearly isn't. It can't even denounce 6 Music as too expensive for such a small audience – not when Radio 3 escapes unscathed with 1.87 million listeners and a budget of £51m.

No more meaningless soundbites, BBC. We know your answer. Now show us the maths.